Transgression and Clifford algebras

Rudolf Philippe Rohr[1]

  • [1] University of Geneva Department of Mathematics 2-4 rue du Lièvre, c.p. 64 1211 Geneva 4 (Suisse)

Annales de l’institut Fourier (2009)

  • Volume: 59, Issue: 4, page 1337-1358
  • ISSN: 0373-0956

Abstract

top
Let W be a differential (not necessarily commutative) algebra which carries a free action of a polynomial algebra S P with homogeneous generators p 1 , ⋯ , p r . We show that for W acyclic, the cohomology of the quotient H ( W / < p 1 , ⋯ , p r > ) is isomorphic to a Clifford algebra Cl ( P , B ) , where the (possibly degenerate) bilinear form B depends on W . This observation is an analogue of an old result of Borel in a non-commutative context. As an application, we study the case of W given by the quantized Weil algebra 𝒲 ( 𝔤 ) = U 𝔤 ⊗ Cl 𝔤 for 𝔤 a reductive Lie algebra. The resulting cohomology of the canonical Weil differential gives a Clifford algebra, but the bilinear form vanishes on the space of primitive invariants of the semi-simple part. As an application, we consider the deformed Weil differential (following Freed, Hopkins and Teleman ).

How to cite

top

Rohr, Rudolf Philippe. "Transgression and Clifford algebras." Annales de l’institut Fourier 59.4 (2009): 1337-1358. <http://eudml.org/doc/10430>.

@article{Rohr2009,
abstract = {Let $W$ be a differential (not necessarily commutative) algebra which carries a free action of a polynomial algebra $SP$ with homogeneous generators $p_1, \dots , p_r$. We show that for $W$ acyclic, the cohomology of the quotient $H(W/$$&lt;\!p_1, \dots , p_r\!&gt;)$ is isomorphic to a Clifford algebra $\text\{Cl\}(P,B)$, where the (possibly degenerate) bilinear form $B$ depends on $W$. This observation is an analogue of an old result of Borel in a non-commutative context. As an application, we study the case of $W$ given by the quantized Weil algebra $\{\mathcal\{W\}(\mathfrak\{g\})\} = U\mathfrak\{g\} \otimes \text\{Cl\}\mathfrak\{g\}$ for $\mathfrak\{g\}$ a reductive Lie algebra. The resulting cohomology of the canonical Weil differential gives a Clifford algebra, but the bilinear form vanishes on the space of primitive invariants of the semi-simple part. As an application, we consider the deformed Weil differential (following Freed, Hopkins and Teleman ).},
affiliation = {University of Geneva Department of Mathematics 2-4 rue du Lièvre, c.p. 64 1211 Geneva 4 (Suisse)},
author = {Rohr, Rudolf Philippe},
journal = {Annales de l’institut Fourier},
keywords = {Lie algebras; Weil algebras; quantized Weil algebras; Clifford algebras; Transgression; classical and quantized Weil algebras; transgression in spectral sequence},
language = {eng},
number = {4},
pages = {1337-1358},
publisher = {Association des Annales de l’institut Fourier},
title = {Transgression and Clifford algebras},
url = {http://eudml.org/doc/10430},
volume = {59},
year = {2009},
}

TY - JOUR
AU - Rohr, Rudolf Philippe
TI - Transgression and Clifford algebras
JO - Annales de l’institut Fourier
PY - 2009
PB - Association des Annales de l’institut Fourier
VL - 59
IS - 4
SP - 1337
EP - 1358
AB - Let $W$ be a differential (not necessarily commutative) algebra which carries a free action of a polynomial algebra $SP$ with homogeneous generators $p_1, \dots , p_r$. We show that for $W$ acyclic, the cohomology of the quotient $H(W/$$&lt;\!p_1, \dots , p_r\!&gt;)$ is isomorphic to a Clifford algebra $\text{Cl}(P,B)$, where the (possibly degenerate) bilinear form $B$ depends on $W$. This observation is an analogue of an old result of Borel in a non-commutative context. As an application, we study the case of $W$ given by the quantized Weil algebra ${\mathcal{W}(\mathfrak{g})} = U\mathfrak{g} \otimes \text{Cl}\mathfrak{g}$ for $\mathfrak{g}$ a reductive Lie algebra. The resulting cohomology of the canonical Weil differential gives a Clifford algebra, but the bilinear form vanishes on the space of primitive invariants of the semi-simple part. As an application, we consider the deformed Weil differential (following Freed, Hopkins and Teleman ).
LA - eng
KW - Lie algebras; Weil algebras; quantized Weil algebras; Clifford algebras; Transgression; classical and quantized Weil algebras; transgression in spectral sequence
UR - http://eudml.org/doc/10430
ER -

References

top
  1. A. Alekseev, E. Meinrenken, The non-commutative Weil algebra, Invent. math. 139 (2000), 135-172 Zbl0945.57017MR1728878
  2. A. Alekseev, E. Meinrenken, Lie theory and the Chern-Weil homomorphism, Ann. Scient. Éc. Norm. Sup. 38 (2005), 303-308 Zbl1105.17015MR2144989
  3. Y. Bazlov, Exterior powers of the adjoint representation of a simple Lie algebra, (2003) 
  4. A. Borel, Sur la cohomologie des espaces fibres principaux et des espaces homogènes de groupes de Lie compacts, Ann. of Math. 57 (1953), 115-207 Zbl0052.40001MR51508
  5. C. Chevalley, Invariants of Finite Groups Generated by Reflections, American Journal of Mathematics 77 (1955), 778-782 Zbl0065.26103MR72877
  6. J. Diximier, Enveloping Algebras, (1977), North-Holland Publishing Company MR498740
  7. M. Duflo, Opérateurs différentiels bi-invariants sur un groupe de Lie, Ann. Sci. École Norm. Sup. (4) 10 (1977), 265-288 Zbl0353.22009MR444841
  8. D. S. Freed, M. J. Hopkins, C. Teleman, Loop Groups and Twisted K-Theory II 
  9. V. W. Guillemin, S. Sternberg, Supersymmetry and Equivariant de Rham Theory, (1999), Springer Zbl0934.55007MR1689252
  10. B. Kostant, Lie Groups Representation on Polynomial Rings, American Journal of Mathematics 85 (1963), 327-404 Zbl0124.26802
  11. B. Kostant, Clifford algebra analogue of the Hopf-Koszul-Samelson theorem, the ρ -decomposition C ( 𝔤 ) = End V ρ ⊗ C ( P), and the 𝔤 -module structure of ⋀ 𝔤 , Adv. in Math. 125 (1997), 275-350 Zbl0882.17002MR1434113
  12. S. Lang, Algebra, (1993), Addison-Wesley Zbl0848.13001MR197234
  13. J. McCleary, A User’s Guide to Spectral Sequences, (2001), Cambridge University Press Zbl0959.55001MR1793722
  14. Nicolas Bourbaki, Lie groups and Lie algebras, (1989), Springer-Verlag Zbl0672.22001MR979493
  15. Nicolas Bourbaki, Algebra II, Chapter 7, (1990), Springer-Verlag 
  16. E. H. Spanier, Algebraic Topology, (1966), McGraw-Hill Zbl0145.43303MR210112

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.